Wednesday, April 10, 2013
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Andrew Wiles, the man who proved Fermat's Last Theorem, is 60 tomorrow. Congratulations.

In 1996, Simon Singh produced this 50-minute documentary for the BBC Horizon series. And I just watched it again and liked it a lot.

Wiles' father was a chaplain in Cambridge – and a professor of divinity at Oxford. The program convinced me – without saying it – that this religious background was important for Wiles to know how to get concentrated for the years that were needed to complete the proof. Note that he first presented a proof in 1993, one that was seen to have a gap, and he was able to fill the gap within 15 months.

Singh's documentary is composed of testimonies of mathematicians. I personally know Barry Mazur – as a kind companion from the Harvard Society of Fellows – and he was clearly a pretty important puppet master behind some key developments although this modest man wouldn't claim credit for that.

Recall that Fermat conjectured, among other things, that\[

x^n+y^n=z^n,\quad \{x,y,z,(n-2)\}\subseteq \NN^+

\] has no solutions. If the exponent were \(n=1\), there would surely be solutions. For example, you may remember from your college that \(1+1=2\). Similarly, there are lots of solutions for \(n=2\); we know them as simple examples of the Pythagorean theorem. For example, \(7^2+24^2=25^2\) if I avoid the two most notorious examples.

But for third powers and higher powers, the identity just can't hold, we know today and people have suspected for centuries. It can't hold for any positive integers. Equivalently (as you may see by multiplying the equation by a common denominator), it can't hold for any positive rational numbers. Most of the famous mathematicians in recent centuries focused on the problem. At most, they achieved partial results, most typically a proof for some value of \(n\).

Note that it's enough to prove the theorem for \(n=4\) – which is particularly simple – and for prime integers \(n\gt 2\). It's because for other numbers factorized as \(n=pq\), one may see that a counterexample with the exponent \(n\) would also imply the existence of counterexamples with the exponents \(p,q\). Those exist for \(n=2\) which is why \(n=2\times 2\) requires a special treatment but otherwise the pattern simplifies things in the most natural way you can think of.

By the early 1990s, the proof had been known for \(n\) up to a very large value. But no general proof existed. In fact, the conjecture was even closer to "fringe maths" in the 1970s when the impatient, industrialized world of state-funded mathematics nearly decided that there couldn't be a proof of the general theorem because the tons of state-funded mathematicians would have already found it.

Well, that was a wrong expectation. The proof existed and was ultimately found by Wiles but it required some modern mathematical techniques that were probably unavailable to Fermat. It seems virtually impossible for Fermat to possess a proof that is nearly equivalent to Wiles': the type of mathematical technology these two men could use differed as much as nuclear reactors differ from steam engines. You just don't expect James Watt to play with similar devices as Robert Oppenheimer.

However, it's somewhat more imaginable – although still insanely sounding - that Fermat had a more elementary proof, one which remains unknown to us. Most likely, Fermat either made a mistake or he deliberately wanted to present himself as a super-genius by a false claim that "he has a wonderful proof that doesn't fit to this small margin". However, he didn't quite fool us because we have rather good reasons to think that Fermat didn't have a proof. ;-)

In the 1970s, the conjecture was returning to mainstream maths because it was realized that Fermat's Last Theorem followed from a modularity theorem ("Taniyama-Shimura"). This theorem says that every elliptic curve – a torus written using complex variables as\[

y^2 = x^3+ax+b

\] and I may publish a crash course in F-theory where I explain these matters in a near future – is a modular curve, one written using the \(j\)-invariant and enjoying lots of special mathematical properties.

These elliptic curves were relevant for the validity of Fermat's Last ex-Conjecture because if there were a counterexample to Fermat's negative claim, you could also construct an associated elliptic curve that isn't modular (weird!), in contradiction with the modularity theorem. These ideas provided the mathematicians with a "sketch" of proofs and the actual proofs were gradually found, by the 1980s. It was firmly proven that the modularity conjecture, if true, implied Fermat's Last Theorem, and the so-called \(\varepsilon\)-theorem was one of the last pieces needed to establish these links.

So the remaining task – one that Wiles solved – was to prove the modularity conjecture for some curves (equations). He did so. The original strategy was to "count and match" the elliptic curves and the modular forms. The minimal implementation of this strategy didn't work so he decided to match the Galois representations instead.

At the general level, the proof of the Riemann Hypothesis will follow the same strategy. A counterexample to the Riemann Hypothesis – a non-trivial root of the zeta-function away from the critical axis – would probably allow you to construct some weird mathematical object that can't exist, either. We may even know what the next step – the rough type of this "weird object" – is. It's probably a weird, non-real eigenvalue of a Hermitian operator or (my strategy) a non-existent representation of \(SL(2,\ZZ)\). Still, we need to know the third step to be closer to a solution.

I've spent hundreds of hours with the Riemann Hypothesis in my life and yes, I repeatedly thought that I essentially had it. ;-)

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Shannon
said...

Impressive.

(This type of mathematical research always reminds me of sailing: when you are facing the wind you must tack (zig-zag) keeping the wind in a 45 degree axis which creates a depression behind the sail and pulls the boat. That's what Wiles had to do by using other new formula and modified problem to find the solution)(The fact that he was working from home must have been difficult for his wife... keeping the kids quiet is mission impossible )(PS. I hate sailing.)

Awesome Lubos, its a good time to review some of this stuff since I have a much better foundation to learn about Wiles' proof now than the last time I tried, I think. That F-theory crash course sounds awesome, I definitely encourage you to write that! I would eat it up.

It's interesting that Andrew Wiles looked incredibly virile, a fine physical specimen of a sporty man as an undergraduate. But twenty years later at the time of his discovery, he came across as physically, emotionally, and mentally feeble. Working in solitary self-confinement on a problem like that must end up causing some sort of decline in one's general health. Yet we only have to look at Ed Witten and see he appears to still be bursting with vitality.

I guess the moral of the story is that if you're going to work on your own, then be prepared for the damage to one's health.

I think that your intuition about the character of this problem is completely wrong. The Riemann zeta function is intrinsically the partition sum in a physical system, the roots of the zeta function correspond to eigenvalues of an operator, and it primarily takes a physicist's intuition to find the appropriate physical system with these properties and look at it in various ways to solve the problem.

Do you think RH will be the last of the Millenium Problems to be resolved? Some experts say it would be much harder to prove P vs NP but fewer people have been working on it and it is much more recent thant RH...

Andrew Wiles devoted much of his career to proving Fermat's Last Theorem, a challenge that perplexed the best minds in mathematics for 300 years. In 1993, he made front-page headlines when he announced a proof of the problem, but this was not the end of the story; an error in his calculation jeopardized his life's work. In this interview, Wiles recounts how he came to terms with the mistake, and eventually went on to achieve his life's ambition.Andrew Wiles on Solving Fermat

There's GOTTA be an easy proof of FLT :-), and I don't mean involving anything like the zeros of the zeta function. Anything else would be an insult to human reason. :-)

Back in the 1980s, I knew an elderly homeless Jewish guy who had no more than high-school math, if that, who would sit in a fast-food place every night trying to discover a pattern in the primes. He'd ask me for computer print-outs of the first few thousand primes in various arrangements, and I'd provide them. He'd then mark up my sheets with numerous lines of several different colors connecting various numbers. He died of a heart attack before he found the pattern.

I'm not suggesting that FLT could be proved by anything so simple-minded as that, but I'd be horribly disappointed to learn that it could not be proved by someone with only two semesters of number theory. :-)

But otherwise he seems to be a very nice gentle person who just loves his work and is able to concentrate on it. This looks not like emotionally or mentelly feable to me, rather he seems to be very modest.

Don't you think that some of the comments in this thread were made by internet bots? Maybe a good idea to have some mild checking of "humanness" for posters.

About what Fermat's own "proof" might have been like: I think the most likely possibility was that Fermat found something like Kummer's proof for regular primeshttp://fermatslasttheorem.blogspot.com/2006/06/fermats-last-theorem-kummers-proof-for.htmland believed that he had proved the general case. (The argument is just a little too long to fit into a book margin).